Index theory is a mathematical framework that connects the properties of differential operators to topological features of manifolds. It primarily deals with the concept of the index of an operator, which measures the difference between the dimensions of its kernel and cokernel. This theory has significant applications in various fields, including geometry, physics, and engineering. One of the most notable results in index theory is the Atiyah-Singer Index Theorem. This theorem provides a way to compute the index of elliptic operators on manifolds and establishes a deep relationship between analysis and topology. Index theory has influenced many areas, including quantum field theory and string theory.